Method for active lap deformation calculation and robot end-effector pose derivation in large off-axis aspherical mirror manufacturing

This paper proposes a deformation calculation method for active polishing discs based on sampling point coordinate transformation, aiming to address the challenge of deformation computation in large-aperture off-axis aspheric mirror processing. The study focuses on the application scenario of robot-mounted active polishing discs for machining large off-axis aspheric mirrors. The key contributions include: (1) Establishing spatial relationships among multiple coordinate systems (parent, workpiece, tool, and polishing disc coordinate systems) and deriving deformation at arbitrary sampling points on the disc surface using homogeneous coordinate transformation; (2) Deriving Euler angle pose parameters for the robot tool coordinate system at arbitrary machining positions; (3) Developing a mapping model between encoder readings and the actual rotation angle of the polishing disc. Simulation results demonstrate that when machining an off-axis parabolic workpiece (F=5.45, off-axis distance=3300mm, aperture=1000mm), the residual surface error between the simulated and ideal surfaces achieves an RMS value below 1 μm and a PV value under 5μm. Experimental results on a 1.1-meter-diameter axisymmetric parabolic mirror (F=5.45, aperture=1100mm) show that the surface form error converges from an initial PV of 52.66μm to 4.61μm and RMS from 6.14μm to 0.88μm, with the workpiece profile steadily approaching the ideal curve. These results validate the effectiveness of the proposed method in lap deformation control. Compared to traditional approaches, this method eliminates the constraints of axisymmetric scenarios and avoids the complexity of high-order fitting, providing a theoretical foundation for efficient and precise machining of large-aperture aspheric surfaces.